Rigged Blender model doesn't preserve volume when exported as FBX
It's a weighting issue: the cylinder is weighted to the cylinder bone AND to the grip bone, while it should be weighted to the cylinder bone only (the same applyes to the trigger and the Thingy). 

You can remove thoose unwanted weights, but as the animation has been done with them, it will have to be redone or twicked.

The difference between the blend file and the FBX probably has to do with the normalization between weights that Blender does as default, while the FBX format breaks.